Amazing lady, confusing Chinese

Hey guys - I am part of a group that uses ESL to help internationals in the US. My primary focus is on Chinese residents in my area, and while I've had students from 8 years of age to, well, old (*grin*), I must say I love working with seniors the most. One lady, however, took the prize for my oldest student ever. I met her last night for the first time, and she's all of 93 years young and healthy as can be.

This lady is from Taiwan, and speaks English, Taiwanese, Mandarin, Japanese, and maybe some others as well. When she heard me using some Chinese with a student, however, the conversation she started threw me for a bit of a loop. Here's how it went:

My problem is this: I've heard people ask ”你会说……？“ (ni3 hui4 shuo1...?), but never just "你会……？“ (ni3 hui4...?). I've also heard people refer to Chinese as 普通话 (pu3 tong1 hua4), 中文 (zhong1 wen2), and 华语 (hua2 yu3), but never 中国话 (zhong1 guo3 hua4). I understand the meaning behind the phrase 中国话, but am wondering if anyone else has heard Chinese referred to as "China language." I'm also wondering if the 说 is necessary or not when asking of someone can speak a language (I always learned it was). Thanks to anyone who can help!